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The Boldgach Conjecture

December 2009

Ask your audience if anyone has heard of the Boldgach Conjecture. Regardless of the response, shuffle a deck of cards and then deal out twelve cards into a pile, while continuing, "It's a famous open problem in mathematics. It says that the sum of any two prime numbers is always an even number greater than two. Many of the world's leading mathematicians—in fact all of them who are greater than two years of age—believe that the Boldgach Conjecture is true. Certainly, nobody has ever found a counterexample. Let me demonstrate with these cards."

Pick up the packet of twelve and give them to a volunteer; the rest of the deck is set aside for now. "Please cut these cards as many times as you wish," you say invitingly, before taking the packet back. "We'll need to split these equally," continue, as you count out half of them into a pile. "Six for you, half a dozen for me! Hold your six cards face down in one hand like my half dozen, ready for transfers."

"You're going to determine how to eliminate five pairs of cards, one by one, and we'll see what pair remains at the end, and whether it supports the Boldgach Conjecture. Each of us now has a packet of six cards. To start with five cards in total must be transferred, some in your packet and the rest in mine. Please pick any number between one and five inclusive, and transfer that many cards from the top to the bottom of your pile, one by one."

If two cards are transferred by the volunteer, you transfer three cards in the same way in your own pile. If the volunteer transfers four cards, you transfer just one; what is important is that overall five cards are transferred. Now each of you places your top card face down on the table, to form one eliminated pair.

"Each of us now has a packet of five cards. So this time, four cards in total must be transferred, some in your packet and the rest in mine. Please pick any number between one and four inclusive, and transfer that many cards from the top to the bottom of your pile, one by one."

If one card is transferred by the volunteer, you transfer three, whereas if the volunteer transfers four cards, you transfer none. Once more, at the conclusion of these transfers, each of you places your top card face down on the table, to form a second eliminated pair.

Continue in this fashion, the numbers decreasing one at a time until each of you has just two cards in your respective packet. One is transferred from the top to the bottom of one of the packets, and the top card of each packet set aside face down to form a fifth eliminated pair.

"We each have one card left: they form the special pair which your decisions determined. I bet that the values represented are both prime numbers, and that those numbers add up to an even number larger than two!" Have the two remaining cards turned face up, they might be a 3 and a 7, or a 5 and a King (value 13), among other possibilities. In all cases, your prediction is seen to have come true. Gather up the other five pairs without comment and put them at the bottom of the deck.

Doable Matching

For a repeat performance with a twist—or simply to redeem yourself with a mathematically savvy audience which may not have been overly impressed with the above—exclaim, "Did I say Boldgach Conjecture? I'm sorry, I think I got that a bit mixed up. I meant Goldbach. The Goldbach Conjecture says that every even number bigger than two is the sum of two primes. There's a subtle difference when compared with what I said earlier. Let me demonstrate with this deck of cards."

Once again count out twelve cards into a pile, and have it cut several times. Deal out six cards for a volunteer, retaining half a dozen for yourself. Exactly as before, have five pairs eliminated, according to decisions made entirely by the volunteer, until just one pair remains. "We suspect from earlier that those two cards have odd prime values which add up to an even number, thus confirming the Boldgach Conjecture. Let's test the Goldbach Conjecture instead: please look at both cards without showing me, and just tell me what the total value is." Pause until the sum is announced.

"Sixteen, you say? There are many ways to get to sweet sixteen by simply adding, but I'm betting that this time it's five plus eleven. You have a 5 and a Jack. That's what I call hard evidence that the Goldbach Conjecture is true!"

Your audience should be reasonably impressed at this point, since those are indeed the cards in question, despite the fact that there are many ways for two umbers to add up to sixteen; but wait, there's more.

Announce, "If the Goldbach Conjecture is true, then it should work for any pair." For the first time, draw attention to the other five pairs set aside. One by one, have the sums of those values announced by the volunteer. Each time, you successfully deduce what the two card values are. (If you don't think it's pure overkill, you can also announce the exact suits.)

The Last Two Cards Match

The above effects are (unnecessarily) contorted manifestations of the wonderful principle which Howard Adams published in 1984 in his booklet OICUFESP (for "Oh, I see you have ESP"). Thanks to Martin Gardner for a recent reminder about the classic "Last Two Cards Match" trick found there.

The basic principle is this:

Suppose a packet consisting of A1, A2, A3, ..., AN, B1, B2, B3, ..., BN,
in sequence is cut several times, and then N cards are counted out---
thus reversing their order---to yield two packets of equal size. Then ifN-1 cards in total are transferred from the top to the bottom of the two
packets (any number being transferred in one packet, and the rest being
transferred in the other), the resulting top cards of the two packets
match in the sense that they are AK and BK for some K between 1 and N.

(It's easy and instructive to check why this works.)

After this pair has been set aside—without being inspected yet—the principle and/of induction ensures that all subsequent pairs down to the last one will match too. The audience has no reason to expect any matches at all, so the revelation of the final match is generally an attention getter in itself.

The standard incarnation of this principle uses identical card values for each K between 1 and N, e.g., with the AK's all Red and the BK's all Black too, and also packages the counting and transferring as a spelling routine. Starting with five pairs instead of the six we've suggested, the phrase "Last Two Cards Match" itself comes in handy there. (Of course, "Those Last Two..." works with six pairs.) It is customary to save as a kicker the revelation that all discarded pairs match.

All we have done is turned things around and arranged it so that the pairs match in a completely different sense. A simple possibility not explored above would be to have all pairs total some constant, such as lucky 13, and work that value into the patter.

For the Boldgach Conjecture effect, all that is necessary is that the twelve cards uses are all odd prime values; simply "cast out nines" and not much can go wrong. For the Goldbach Conjecture effect one could memorize and use these cards:

The usual considerations are in force with regard to "shuffling"—maintaining the top third of the deck undisturbed, and so on. Suit memorization is optional; note that we have matched royalty with gems.

The vertical pair totals shown above are 4, 6, 12, 14, 16, and 20. The 2's are included for fun: when a total of four is reported you can muse that, "As 1 is not considered to be a prime, the cards can't be an Ace and a 3, and so must both be 2's, which are prime, oddly enough, once again confirming the Goldbach Conjecture."

There is another advantage to going through the whole transferring routine twice as suggested above: it's easy the first time around for the volunteer to get confused and misstep. It won't matter in the first routine, but the practice (and your keeping an eye on possible trouble spots) should ensure perfection for the more sensitive second effect.